 Now, I'm really delighted to kick off the evening by introducing a longtime friend and supporter of the museum, Mr. Neil Chris. Good evening. Every good speaker needs a good introduction, so that's my job here, and I'll take approximately five minutes of your time. So first of all, I want to thank the Museum of Math, the Simons Foundation, and one of their board members, Manjul Balgava, who invited me to give this introduction, and thank you to the Museum for organizing these talks. My name is Neil Chris. I was asked to give a brief introduction to today's talk. You can see the title there, Spacetime and the Fourth Dimension by our very distinguished speaker, Robert Degraff. I think you're in for a real treat. In fact, I know because I sat through the first talk this afternoon, because, you know, not only it is a terrific topic, but it's being discussed by an equally terrific speaker. Robert is eminently qualified to give this speech for several reasons. Number one, he is a mathematical physicist, and the talks about the geometry of spacetime, which is a very mathematical and physicsy topic. It's about deep questions in physics, the Big Bang Theory, black holes, quantum mechanics, and so on, and he will be covering all those things. Number two, Robert is and has been the director of the Institute for Advanced Study, an academic institution, right over in Princeton, New Jersey, which traces its lineage back to 1930, when Albert Einstein, who is a major figure in this talk, was one of the first professors. Robert leads this prestigious organization, and it has four schools, mathematics, natural sciences, where physics takes place, historical studies, and social sciences, and he is not just the director, by the way, but he's also the Leon Levy professor of the Institute. Further, as a mathematical physicist, he made significant contributions to the field of string theory, so you can trust whatever he's going to tell you tonight. In fact, as an aside, a little about me, IAS is where I met Robert. I've been a trustee at the Institute since 2011, and back then I was a member, back in 1994, I was a member of the School of Mathematics when I was actually an academic mathematician, which I think is how I secured this gig tonight. Lastly, the third and final reason Robert is such a qualified person to deliver this speech is his enormous, almost second career as an exposer of science, author of popular books about science, and television personality, so far as I know. Robert is actually considered no less than the voice of science in his native Netherlands, where, by the way, he was president of the Royal Netherlands Academy of Arts and Sciences from 2008 to 2011 before he joined the Institute for Advanced Study, and in doing a little preparatory research for this talk, I recently discovered the depths of his fame when I learned that there is a website called RobertDigraph.com, which lists various books and speeches that he gave, including a Dutch book called De Juniore Bethe Canon, which means in English, the junior beta canon, and honestly, I tried to find a different translation, and that's what it is. But I think this is funny, and then sadly it's less funny because it actually turns out Robert has an explanation which I'll let him tell you, but the website has this to say about that book, and it really actually sets the stage quite nicely for tonight's talk. The blurb about the book, which may have been translated from the original Dutch, says Dutch famous scientist Robert Digraph discusses in his junior canon in an original and accessible way the main topics of science, like why the toilet for humans is such an important discovery, or who was Newton, and what is the Big Bang? So those three important questions. Toilets, Newton, Big Bang. So with that, I'm going to turn it over to Robert, and I hopefully he will discuss these topics gracefully as best he can. Thank you. Thank you, Neil, and thank all of you for being here and spending the evening with math and complicated matters in physics and the universe. Coming back to that story, in fact, it was a book for children where we had 50 topics that everybody in science should know, and we had great fun that our first topic was zero, because that's where all mathematics began, but then there was actually that year. There was a big debate among biomedical researchers what was the biggest invention effort to save lives, and the conclusion by far number one was the toilet. I think number two was antibiotics or something, but hygiene was saved more lives, I think, than the combined efforts of science together. So that puts us in the right context. But we will cover the Big Bang, Black Holes, and perhaps a little bit of zero here, too. My topic will be about space, time, and the fourth dimension, and I hope if I achieve one thing that after this lecture, you will think differently about time. And not only that you wasted an hour, but that you, in some sense, have a good sense. And I want to start with this famous book, Flatland. Perhaps most of you have seen it, or Modern Incarnation. There's an even animation movie about it. It's written by a math teacher in Victorian England. Trying to describe the concept of extra dimensions. By assuming there's a world that only knows two dimensions. And I feel I'm eminently qualified to give this talk because I'm Dutch, and I think the Netherlands is the flatest of countries. So a very two-dimensional perspective on the world. It's interesting for many respects also because it gives you an insight, not only in mathematics, but how Victorian England works. And it's a very hierarchical society. Most Flatlanders are triangles. And if you are a polygon, the number of edges gives you rank as society. So you can be a square. I see many pentagons here in the room. And the highest order is if you're a priest. And then you have an infinite number of vertices. So you're the circle. But the remarkable thing is you also describe there are two guns, which are line segments. So even if you look from the wrong perspective, you wouldn't even see them. And these are the women in Flatland society. So you see there was still a lot of to be gained. So there's the men's door, there's the women's door. Everett has great fun using these kind of exploring these gender inequalities. But there's something that happens in Flatland where certainly things can appear in the middle of the room. And they don't go to any door, neither the men's nor the women's door, and come and disappear and create great turmoil in Flatland. And this thing they call the sphere. And you probably understand what it is. There's a third dimension. So sometimes the sphere comes from the third erection, which is totally oblivious to Flatlanders. Now I think once a biologist told me that ants, they only have a sense of two dimensions. So whenever you hit an ant, it probably will come from a dimension that the ant isn't aware of. They only have a two-dimensional sense of space. But I want to basically convey to you that we are all Flatlanders. And this funny thing that comes out of the hidden dimension is around us all the time, because this extra dimension is time. Now to get used to a little bit about dimensions, now mathematicians can work in arbitrary dimensions. So we think of a cube as something in three dimensions, but you can do it in any, like it's zero dimensions, only one point. In one dimension it's difficult to see, you just get a line segment. A square is the equivalent of a cube in two dimensions. Here you see the three-dimensional cube. And here's a projection of what's called a hypercube, a four-dimensional cube. And you can go on like five, six, ten, you can draw cubes in a million dimensions. But how do we think about this? So here you see a two-dimensional object. And this is, well, what actually are you seeing? Well, first of all, I hope you admire my PowerPoint animations. Everything is homegrown tonight. So you might think, well, this is a three-dimensional object. But of course it's not, you know, it's a screen. You're looking at the screen. But even if I had constructed a real three-dimensional cube, well, I actually have one right here. This is a three-dimensional cube, but actually what you are watching right now is your retina, the backside of your eye, which is totally flat. So if I move this thing, your brain is looking at this sequence of flat images, two-dimensional images, and reconstructs the third dimension. The third dimension you can't see, you can only reconstruct it in your mind. So why not stop there? And so for instance, here you see just as you can project the three-dimensional space in two dimensions, you can project a four-dimensional object in three dimensions. This is a projection of a four-dimensional hypercube that's moving around. Now I had a colleague, a biophysicist at NYU, that for a full year, looked and manipulated these four-dimensional images in the hope that his brain would catch up and that he would see the fourth dimension. Now I'm looking for volunteers, because one year was not enough. Nothing kind of clicked. But it's, you know, I hate to suggest, but perhaps we know we have children that grow up in fourth dimensions. In some sense, you might be able to see. My favorite story is the famous mathematician Bill Thurston once gave a talk. He was one of the few people who could think in higher dimensions. He gave a talk about mathematics in four dimensions, and there was an audience like this, all professional mathematicians, and he was moving around and nobody could follow him. At some point there were all these blank stairs, and then Thurston looked into the audience, oh, this is clearly not the right way to present the material. You shouldn't think of this in four dimensions. You should do it in five dimensions. But it all became simple. So only very few, I think, can actually get some intuition. But I will try to explain that we do have an intuition for the fourth dimension, but it's in some sense not connected to space but to time. Now there's another way to think about higher-dimensional objects. For instance, you can think of an instruction to fold a cube. So this is something that flatlanders could understand. They could cut out. But when you say the flatlanders connect, glue all the edges together, they would look at you, well, that's clearly impossible. But then in three dimensions you could do this. This is the analogous of this kind of construction package for a hypercube. So you can take this object here, and if you are told to glue these ends together, you cannot in three dimensions. But in four dimensions you can do it. You get a perfectly symmetrical object. So this is one way to think about the hypercube. Some of you know perhaps from a famous painting by Salvador Dali, very popular in the 1970s, I think. Many students had this poster. A mathematical poster. So in fact, there's something, just as a little aside, I want to tell you, mathematicians can work in arbitrary dimensions, but if you would plot the difficulty of doing math as a function of the dimension, it looks something like this. Dimension one and two are very simple. Three and four are complicated. And then in some funny way, if you go beyond dimension four, things become simpler again. Now this to say there are lots of mathematical results that were first proven for dimensions five and higher. I don't want to go into the technicalities, but there's a thing called the Poincare conjecture. But it's some way to see whether a space has no holes. It's essentially a sphere which was first proven by Stephen Smale for dimension five and higher, and then dimension three and four took a long time. Actually, the proven dimension three was only in the 2000s by Gregory Perlman, and it was a fascinating story. But again, it shows that space, three and four are special dimensions from a mathematical point of view. If you ask mathematicians why four is special, they would say, well, if you have a surface, and you would have a surface in moving in four dimensions, because of the equations two plus two is four, it could self-intersect. So there could be a way just if you draw a curve on a plane, and you can have an intersection, the same way if a surface is moving in four dimensions, and if I have time, I will say something that there are some physical theories that point out that perhaps this is the reason why we live in four dimensions, and why we live in four dimensions will become clear in a moment. Another thing that you get used to, this is a knot, knots are lines, actually circles, embedded in three-dimensional space. It's interesting to see whether you can a knot, a knot, or not. But actually this only works in three dimensions. So if you see what is the problem in untying a knot is that you have two strands, like the red one and the green one, and this is just a slice through the knot, and you will basically pull the strand through one through the other, and you kind of do, because you're not allowed to break the line. However, you have an extra dimension, say a fourth dimension, you could just kind of hop over. It's the same like in flatlanders, they would have borders, but if you are a three-dimensional person, you could like fly over the border. You wouldn't be bothered by any walls in a two-dimensional sense. So dimension three and four are special. There's one more reason I want to point out that dimension three and four are special. These are the so-called platonic solids. I have a few here. The Greeks were already fascinated by them. These are five regular objects. The first three, the tetrahedron, the cube and the octahedron, exist in any space-time dimension. I already showed you cubes in dimension four, five, six, so that's very simple. These two guys, the icosahedron and the dodecahedron, are special because they only exist, they're made out of pentagons, 12 pentagons or 20 triangles, they're kind of closely connected. They only exist in dimension three, and these exceptional objects, they do not occur in any other dimension, except also dimension four. So if you would be a four-dimensional geometer, you would know about these three objects. They are the so-called 24 cell, 120 cell and 600 cells, pretty complicated objects. If you can see them, in some sense, it's gluing, I think, 600 of these dodecahedron together, and I'm afraid none of us have 40 vision. But if you would have 40 vision, you would kind of understand why this actually fits together and why it's beautiful. I must say, by the way, this dodecahedron is very special, especially for the Greeks. So Plato writes down what I like to joke about of elementary particles. The Greeks knew there were four elements, earth, fire, water and air, and they also knew there were five pentagons. So their theory of everything was that you can associate an element, a figure, to each of the four platonic solids. So earth, obviously, is a cube, because only cubes you can put together and make a solid out. Fire, you can burn yourself, so it must be pointy and sharp, so that's why it was in triangle. Water rolls, it's very fluid, so it had to be the ocahedron. I'm not sure what the theory about air was, but then there is one left, and then Plato says, perhaps this is the element out of which the planets, the stars, the heavens are being made. I kind of can sympathize with it. In many ways I feel this is the most beautiful of platonic solids. I once was a photographer who picked this one. I said, why did you pick this one? He said, well, it's complicated, but not too complicated, which is a good capture of mathematical beauty, I think. So it's interesting if you follow art history, you see this fifth element, quintessence, which is symbol for a metaphysical symbol, a symbol for astrology in many different paintings. And in fact I want to show you this. This is the old parliamentary hall in the Netherlands, in The Hague, and it has this beautiful dodecahedra as kind of light fixtures. And so when this was opened, and there's an artist who made this, and there's a lot of vibrations, resonances, with thousands of years of history, there was a commentary in the Dutch newspaper saying it was very inappropriate to have this beautiful hall and then have two soccer balls to... So we have to revive that thing. But so much about dimension three and four, and now I want to kind of bring in time. So I will start with space, certainly I think in the old days, so I will come in the moment in more complicated ways to think about space, but space essentially was seen as the stage on which physical objects move and behave. And Euclidean space is rigid, it's infinite, it's flat, it's this kind of perfect frame where things can happen. Similarly, time, in Newton's words, is true and mathematical and absolute, so time is the big kind of conductor that makes sure that all physical phenomena happen in a coordinated fashion. As it said, the famous line, time was invented to prevent everything happening at the same time. And then of course there's the famous remark or insight from Albert Einstein that actually time should be seen as the fourth dimension. So in fact, actually it's not Einstein's insight, it came from a great German mathematician, Hermann Minkowski. In fact, there's a very nice relation of Einstein to Minkowski. Einstein was a very bad student in Zurich, not bad in the sense that he wasn't smart, but he was always fighting the professors, etc. And he writes as one professor he admires, which is Minkowski. But he's a brilliant mathematician, but Einstein then says, I do not have time to attend his lectures because I have more important things to do. So it's only after he announces his theory and actually after a few years that he gets convinced that Minkowski's point of view that four dimensions is the way to think about space and time is the right way. So I want to actually try to explain that to you and I want to take again the perspective of flatlanders. So if there's a flatland Einstein, he would be saying flatlanders, there is a third dimension and that's called time. So we only have two space dimensions for a moment. And right now you're looking at a movie and it's a movie of two particles rotating around each other. Now you're not actually looking at a movie. What you're looking at, what you're actually always looking at if you see any movie, is a sequence of images. But they go very fast. So just as your brain imagines the third dimension, so your brain also imagines time because you think it's a continuous movement. It isn't. You have snapshots and you glue them together by interpolating in your mind. So now what I want you to do is think of this motion in time as a sequence of images and think of the separate images, not one after the other, but one on top of the other. So we're going to stack them, what you see here on the left. So these are the images and the earlier one are in the beginning and time is going upwards. So in images I often think about if you have your holiday pictures and you print them out, you get a stack of prints and it will be in chronological order. And this is the beginning of your vacation, this is the end. And so here you can look at, so if you now track one of these particles, you see it actually, it becomes a kind of a line that goes upwards and here on the right-hand side I've glued all the images together and now what's called the space-time continuum. I filled in all the slices. And in some sense what you're doing on the right-hand side is taking space-time and if you would take a big, you cut it through the middle here you would have actually one of the images over there. So unfortunately our mind is not able to process everything at the same time. So you're following me second by second. In fact, you should realize that even if you're sitting perfectly still you're moving in space-time. You can't stop time. I can stop my motion, so this hand is not moving in space now, but it's moving in time. In fact, with a rate of one second per second you can't prevent that. So we are all moving in time now. But in some sense we don't realize it because we always think in the present and so we're always in one of these time slices. But I slowly want you to think about all these slices together, kind of a holistic view of a part of your history. So I love this thing, this Museum of Mathematics, so terrific. It has a time slicer here. So it's over here. So I want to kind of use it for a brief moment. So I showed you... So this could be a trajectory in time, but so this would be kind of looking at it in space. So, you know, you would have just... So it would be a circle. Time is now flowing in this direction. I'm moving it through time. This would actually be quite radical because at some point, you know, it would... What would it do? Well, it would shrink and it would disappear. I can do another thing. Actually, this is a beautiful thing over here. So this is a complicated version of the image you see here on the right-hand side. This has two of these spaghetti strands. Remember, these are two particles orbiting. So it would be like the Earth and the Moon. This would be three of these objects. And you know, perhaps you can see what's happening if I move it through. So does anybody have an idea what this motion would be? It's something that you see sometimes magicians do. Anybody? Juggling, exactly juggling. So if you would think of the space-time trajectory of a juggler, you have this object over here. And then you start slowly to appreciate how nice it is to have space and time connected because you no longer have to think of the motion. See the whole magic trick at once. This is the whole history of the juggler. In fact, you could do other things. For instance, you have to think what are... We know what points in space are, but what are points in space-time? Points in space-time are things which only have a position but have a time moment. So this is an example. The moment I snap my fingers, it's in space and in time. So we call these events. They're even smaller than points because a point is a line in space-time. It's there all the time. This is a point that only exists for the briefest moment of time. I often think about fireflies. If you have fireflies, they go off like this. So a firefly, the flesh only lives for a very brief moment. And so that's a space-time point. In fact, if you literally think about a flash of light, and I want to illustrate this, there's just a flash. The light will travel. It will expand. So think about, perhaps, more easy in a flat land. Think of throwing a pebble into a pond. You have the moment the pebble hits the water, but then the wave will spread. So I can do this here. So this is kind of the moment the pebble or the light enters the pond and here it's spreading. You see? This is the opposite. So if you think of a flash of light that goes off or anything that creates a disturbance, you create this kind of comb which is called a light comb. It's actually, in this case, a two-dimensional object. In our case, it will not be an expanding circle because that's a water wave. It will be an expanding sphere. So the cone will be not a two-dimensional but a three-dimensional object. You should get... So here is a good example. So this is my animation. This is the the pebble in the pond and this is the reverse trajectory. And here you see it now visualized in space-time. This is the space-time point. It's the here and now. And here is the flash of light. Now, the rockle thing is that anything light is the fastest moving object in physics. So anything that is hit or in some sense causally connected to this moment, anything that can happen as a result of that flash happens inside this cone. Because if you're standing still, you go straight. If you go under light in this image, you go under 45 degrees. So the inside of the cone is what physicists call the future. When I first saw it, it was quite exciting because I never had a picture of the future. But you really can see this. In the same way, you can think of the past. The past is anything that happened and influences the future now. So I have a past. A fact that goes all the way to the Big Bang. It is a gigantic light cone. And I have a future. And we have different light cones. Yours, you might sit next to me. So you have a slightly different... And it's this huge space outside the light cone, which is like no causal relation. One of the great figures to have at the Institute for Advanced Studies, Freeman Dyson, a great physicist. But he remembers reading a book when I think it was six or seven about general relativity, about relativity, Einstein's theory. And reading about the future and the past and this what was called the absolute elsewhere. And there is a wonderful cartoon in one of the magazines at that time that I think was conveyed by his father because his father asked him, Freeman, do you know where your sister is? And his answer was, she is in the absolute elsewhere. So that was a brilliant cartoon, I think in the magazine punch. Very, very smart. So it's interesting that most of space-time is in the absolute elsewhere. These are space-time points that you cannot influence. For instance, right now something is happening in Beijing. There's no way what we do now can influence Beijing because at least the time that light has to take to travel there which is a tenth of a second. In fact, if you look at the sun you look at something that is eight minutes ago. So something happening at the sun right now is in the absolute elsewhere compared to us. So you have to start thinking about these space-time pictures. I want to just test your understanding and I have this machine where I go to manual mode. So I draw on this figure. The time is flowing upwards. Does anybody have any idea what this could be? It's a representation of something. Anybody, any idea? The Big Bang. The Big Bang. Close, close. Something a little bit more every day, perhaps. Yes? A hand. No, think about this as time. So you should think of this as a sequence of events. So there is one thing, nothing happening and certainly a lot of parts are flying outwards. Fireworks. Yes, exactly. That's what I was thinking. Explosion, whatever. So Big Bang is fine but there's also something before the Big Bang here. Yeah. Just another one. So again, think of this as moving upward in time. So suppose the two black lines are two people and they're doing something. What are they doing? Anything Yeah, exactly. One throws something to the other. So this could be... I was thinking about a tennis match but it could also be just throwing a ball. So there is here person A throws a ball, there goes the ball then they shoot off. So it could be ping-pong or anything. So you can actually draw anything in terms of these space-time diagrams. It's important to emphasize that for physicists they often think in these kind of space-time diagrams and what's kind of special about it is that it captures all of history at once. Now there's something much more spectacular because right now I've made you think about space-time but it feels like a pre-sliced loaf of bread. You still have these slices which are space and then time is which slice are you. But Einstein said it's not sliced. It's really one thing. You can actually do rotations. You can do funny stuff mixing space and time. In fact, you can slice it in many different ways. For instance, this is a... So here the red and the green are two events. So it could be me clicking my hands like this. So these are simultaneous moments happening in one time slice but if you have an alternative view of the world and you slice things under a certain angle so is this allowed or does it mean? In fact, it turns out that if you move through space with a certain speed you're still slicing but you're slicing at a small angle. Now the red and the green are different because you see the green which is this one happens first and the red happens after that. So the idea of happening at the same time doesn't make sense anymore. In the same way of two points in the plane and I say, well, are they on the same line? Well, you can always draw a line to two points. A space doesn't come pre-sliced and Einstein's big insight where that space-time is also not pre-sliced. In fact, you can slice it anywhere you want. These are nice slices that belong to uniform motion but if you move like crazy then actually the slicing is very crazy too. You can do geometry in this space-time. So you could give a course in space-time geometry just as the Greeks studied three-dimensional or two-dimensional geometry. So you'll know Pythagoras theorem, a squared by b squared is secret if you have a triangle. In this new math which is called Laurentian after Hendrik Lorenz who was a great Dutch physicist who predated Einstein I always think Einstein is the greatest physicist most people think, certainly modern times but I always like to ask the question who did Einstein think was the greatest scientist? And he had a very definite answer, was Hendrik Lorenz. And he wrote many of the equations that Einstein used without daring to give the physical interpretation. So Lorenz worked with space-time but never dared to mix space and time as Einstein did. So in this new geometry you still can draw triangles. You can compute the length of a, b and c. In fact, you don't get a squared with b squared to c squared you get a squared minus b squared to c squared so it's a very different kind of math. It has strange consequences. For instance, one thing we all know, if there are two points and you want to go from one point a to b the shortest path is a straight line. So the dotted line takes you longer it's a longer path than the straight line. In space-time it's exactly the opposite. If you have two events say this one and this one, they can be connected. They can be connected by my hand, it's standing still. You could do something more complicated. You could go in a rocket and move away to some outer galaxy and hurry back just in time for the second click. It would be a very complicated path. If you calculate in this theory of Einstein you find that the curved path is shorter than the straight path so the standing still takes the longest time. So most of you might have heard about the twin paradox. The twin paradox is like these two twins. One stays at home on planet Earth, the other goes into a rocket, makes this grand tour of the universe, comes back and finds that his brother is much older than he. In fact, that experiment has been done. They put two clocks, one on planet Earth, the other in Boeing, that went around the world I think for a few years and then checked the two clocks and then the clock in the airplane was a few nanoseconds younger than the one that stayed in the airport. So that's the kind of bizarre math that you get if you go in this four-dimensional, but the important thing is it's there. We can study it and it's kind of fascinating. But I now want to get you to where it really gets excited. If you take this theory about space and time and combine it with something else, quantum theory, which is the theory of the smallest particles. And in fact, quantum world is very different from the classical world. One way to put this is that in quantum world many things are uncertain. Basically, in the quantum world anything can happen, however crazy it is, but with very, very small probabilities. This was carried home a few years ago when perhaps you remember they switched on the Large Hadron Collider, this big particle accelerator in CERN. And then there was a message in the newspapers, well perhaps they create a black hole that would destroy the Earth and the Universe and they asked one of my colleagues, can this happen? He said, yes, but very small probability. And he meant like 10 to the minus 100 or something. But then people all got excited when the world will disappear and many, I still have the newspaper with the headline, will the world disappear tomorrow? Just the fact that you still have that newspaper is very good. But so the thing is that you should now start to loosen up a bit and not only think of straight lines and fixed lines in space-time, but think about many different ways in which particles or even you are moving in space-time. You don't realize this is a small probability. Some of you might certainly end up outside the museum without going through any wall or something. It's an extremely small probability and I think none of us have ever witnessed this. But particles do it all the time, like a radioactive particle. It's locked up in a cage in which it cannot escape. Yet once every while some escapes against the laws of physics just goes through the wall. So these things do happen if you are an elementary particle. So I actually want to illustrate how the two things are merged together by asking a question, which I think is a very deep question, that perhaps some of you have never thought about it. The question is if you learn in a physics course, say the mass of the electron. Say, well, this isn't this number. Why are you only giving one number while there are so many different electrons? Isn't like, could one be a little bit more heavier or lighter? Perhaps there are electrons that are very old, have lots of scratches, they have eroded. Why is there a whole range? It would be very strange to learn the weight of a human being as one universal number. So is there a factory that builds elementary particles? And is it like the perfect factory? How does this work? So the nice thing is an answer to this and the answer came in a wonderful episode. Again, this is Richard Feynman. Feynman was a graduate student in the early 1940s in Princeton and John Wheeler, famous physicist, was his advisor. Wheeler invented the term black hole. He worked with Niels Bohr in quantum mechanics. He worked on the hydrogen bomb. He basically invented many issues in gravitational physics. And Feynman actually, when he got the Nobel Prize, you have to give a lecture and in that lecture he describes the incident. One of the night, Saturday night, very late at night, his advisor's on the telephone, kind of dubious situation, and says, Richard, have you ever thought why all the electrons are the same? I know the answer. There is only one electron in the universe. So this is his theory. You are now accustomed to spacetime. If I put an electron here, it will go up straight like this. Like you are doing, you are moving in spacetime now. Wheeler had the following idea. He said, okay, suppose I want to make an identical copy of myself. How can I do this? It's very easy. You first build a time machine. You go in the time machine, you end up like five minutes earlier, walk up the stairs, come here, and stand next to yourself. And you know you can do it again and again and again. And you can produce as many copies of yourself. They're perfect copies, because it's you. So Wheeler's idea was suppose that the electrons would go up in time, but are allowed to go down in time again. And up and down, and up and down, and up and down. And weave this kind of big knot in spacetime. An electron going down in time actually would have a physical interpretation. It's what we call an antiparticle, an anti-electron or a positron. A particle with the same mass with the positive charge instead of negative charge of the electron. If you would have this diagram and you think of it as a... I put my kind of big spacetime slicer here and put this whole knot through the slicer. Actually we could almost do this. It's still on. So this would be a slicing one electron. If it can build knots, it can do something like this, right? So it would be have one electron and then at one moment a pair creates out of nothing. It would go. Now we have three. Then these two disappear and have one again. So if I'm able to do something like this, for a brief moment I'm with three particles, two particles and one antiparticle. And this was exactly Wheeler's view. So you start with one, but if you look closely, you see many. And they're all the same. Why? Because they're the same. It's like if you have a big knot and you cut it through, of course the color of the strands are the same because it's all the same knot. Well, apparently Feynman and Milley said if this is true, there should be equal numbers of positrons of electrons and positrons, which clearly is not the case. Well, we're all built out of atoms with ordinary atoms with electrons. So there should be some huge depository of antiparticles somewhere in the universe. But the idea actually is, I would say it's actually true. This is true. If you, to cut to the chase, if you look through a microscope at an electron, what they do for instance in the big particle accelerators and you close up, you see that an electron is not an electron. An electron is actually a big fuzzy cloud of particles and antiparticles. Only at the long distance you see it as one. If you zoom in, you see this whole cloud, which means that an electron viewed on a gigantic microscope with a particle accelerator is, has very different properties than an electron that you, for instance, see and appears in the physics books. For instance, its charge is much higher than it actually is if you observe it at, say, at room temperature. Now, this Feynman dismissed this idea and he didn't return to it. And, you know, he went to Los Alamos, he helped build the atom bomb. He became this physics star. He becomes a professor in Cornell. He's the most brilliant person around. And then nothing comes to him. And he's feeling, oh, I'm failing as a physicist. And then certainly he recalls this conversation with Wheeler. And he starts actually to take that idea seriously. And I like this picture because it's the first page of his notebook where he actually is sending these electrons back into time. And he's doing little calculations with pluses and minuses. And he finds out it works. Actually, it's a beautiful way to describe quantum mechanics. You should actually send particles back in time. And that's a one-way to understand. And that theory is now called Feynman diagrams. So he makes these diagrams. He writes somehow in a letter to his sister. He said, oh, it's so fun. All the other physicists are doing calculations and I'm making drawings. And perhaps at one point, all the physics journals would be full of drawings instead of calculations. And now it's true. Actually, you see these Feynman diagrams everywhere. In fact, you see them even in this way. So this is a famous van in physics. And the canonical anecdote that goes with this van is that a graduate student of physics saw the van in the 1960s in LA. And a woman was driving the van. And he stops the van. He says, madam, I have to tell you that the images on the van, we use them all the time. They actually are physics diagrams. And we call them Feynman diagrams. And then she says, yes, I know. I'm Mrs. Feynman. And here's Richard Feynman and his wife and children with the Feynman van. Actually, the Feynman van has been lovingly restored. Last year we had this centenary of Feynman's birth. Here I'm with the van. And now I think it's bought by some museum. I hope. I mean, it was auctioned off. And so I hope it's... I actually want to paint the Institute van with these diagrams, which would be terrific. So we draw these things all the time. And this is the Feynman diagram I just showed you. The original one that we all see. And now I could ask you what does this stand for, but you already answered the question. I showed you the, say, think of the tennis match that you looked at. So I would say this looks like one tennis player hitting the ball or somebody throwing a ball to another person. So in this case, it's not persons and tennis balls. It's two electrons that are exchanging a light particle. So the modern way to think about physics, physical interactions, electric magnetic interactions and forces is actually these electrons are... It's like a laser game. They're shooting laser pulses. They're shooting electron... Electrons are shooting photons. And this is a very visual way to think about a repelling force, because there's a backlash when you shoot off your gun and then there's a recoil on the other side when you catch the electron. Now, one little exercise, perhaps somebody has an idea. If this is a repelling force, what would be an attractive force? Can you make an attractive force? So you should be pulled together by anybody an idea. Yes? Instead of shooting from electron to electron, you can make electrons to electrons. Ah, you think something with anti-particles. No, two electrons. But you're getting close. So you should think in terms of space and time. So it would be terrific if you think about this person should first get a push. So it would be terrific if you can first catch the ball and then throw the ball. This seems very illogical to do. But remember, we can play with time, etc. So you can have this process here, where in some sense the particle is... You throw it in some sense in one direction and you catch it in the other. This already shows that these photons, these tennis balls, are not behaving like ordinary tennis balls. Not at all. And the technical term of this is virtual particles. So if you ever see these diagrams, you try to take them completely seriously, you cannot in the sense that these intermediate objects, the balls that they throw, particle throw, do not really exist. They exist in the physical interpretation, but you can never measure them. If you, for instance, this would be the image that physicists would draw if they want to describe why two magnets are repelling. Well, you don't see flashes of light going between two magnets, right? But in the diagrams, in the physical interpretation, in quantum mechanics, where things can happen, yet you cannot observe them. Actually, this is the way to do. And you can have even more crazy stuff. For instance, you can have the following. Anybody idea what this process would mean? So I again invite you, this is the electron, to think of this as a sequence in time. What would this be? Think of the explosion that we've seen. I was thinking of a river that bends around something and then joins again. So it's not in space, it's in time, right? So here we have one particle, here we have two. So it's a process where a particle would kind of split in two and then the two would combine again and go back, go through as one. So all of these things happen and you can even have more crazy. So this would be a brief moment where a particle slightly behaves slightly different, that splits into particle and antiparticle and goes through. All these processes can happen and in fact you can measure them. They are measured in experiments. In fact, you can have the following thing, which is totally perfect. This is again something I could illustrate if you really had a balloon like this. So think of this as a process in time now. So here's my time slice. So what would it look like? So we have nothing, right? Then out of nothing, a circle comes, it splits in two. Now I have two circles, then a moment later they combine again and there's nothing. So this is a process where you can think out of nothing, out of empty space, a particle and an antiparticle are created. They live for a brief time and then they hit again and they combine to zero. There's another way to think about it is as a process where literally you have a time machine. This is a particle that goes up and down into time. Now this might be the point where you're saying this is getting really silly. But actually this is a fact. This is happening right now as we sit here. There's a lot of space around us, empty space. In the empty space there's continuous creation of particles and antiparticles. You can measure the energy that they capture. In fact it has a good name. It's called the Casimir energy. It's a physical phenomena that you can measure. And in fact, so empty space, this is my PowerPoint imagination of empty space. It's just like this bubbling pot of particles and antiparticles that are continuously being created and disappearing again. And it's a nice quote by the physicist Murray Gellman that says in quantum mechanics, everything that's allowed is obligatory. So anything that can happen does happen with a small probability. So thinking about this, you only can think about this if you have the space-time picture. So it's absolutely crucial for it. Now in the remaining minutes, I want to say a few words about the next step, which is introducing gravity and thinking about what happens when you think about space-time in an even more general way. And we have to go back to Einstein and now we'll be talking about Einstein's general theory of relativity, which is much more complicated. So you all know the equation equals mc squared. This is equation for the general theory of relativity. It's not that famous. I wish it was. I try to be an advocate for this equation, but it's like I think a little bit too much. But you can think of it as in terms of a slogan. And the slogan is that on the left-hand side is space, or actually space-time, and the right-hand side is mass. And one way to think about it is that the two things are being combined. And so what literally happens, so this is again the piece of flatland, and think now flatland is not made out of a rigid material, but flatland being made out of rubber. So the flatlanders don't notice that their country has certainly become so flexible. But it has enormous consequences because if you put something in flatland, it actually reacts to it because since it's no rubber, it will kind of curve around. So the flatlanders are now moving, they're still living in two dimensions, but now living on a mountain landscape. And they see funny things. For instance, if they want to go straight, what does it mean to go straight? If you would roll a ball, it would start to curve around. So motion would be much more complicated in this. But you can do this. So the curvature actually of space will tell you how to move. And that's actually what Einstein's equation says. It says mass tells how space to curve, space tells how mass to move. This is the two-dimensional version. It's actually not two dimensions, it's three dimensions, of course, which are curved. So you should think of space as a big sponge that you can squeeze and push. In fact, you even have to think about space and time. So you have to think in curved four-dimensional space, which is really, really very difficult, right? The only thing I think we all can think of curved space is because we have an extra dimension. So we already needed a third dimension to think about the curved surface, because then I can think about the mountain valley. But if I want to think of the curvature of four-dimensional space time, it needs a fifth or a sixth dimension, it becomes really, really difficult. And so Einstein was really struggling with this extremely lucky that all the math was already worked out. So in the mid-19th century, Riemann and others, mathematicians, thought about abstract spaces in general dimensions, thinking about curvature and all of that. So sometimes physicists have to invent the mathematics, the language to describe their physics, like Newton had to invent calculus. But actually in this case, Einstein could basically just take the theory from the shelf. There was already a subject of mathematics that went about curved space times. And the fact that space was curved was famously demonstrated exactly 100 years ago, in 1919, when a famous exhibition discovered that the light of stars that go very close to the Sun, nicely illustrated in this thing, actually curves the images of the stars. And these days you can see this actually with gigantic telescopes, it's a beautiful thing. And of course, Einstein now was able to do something which is even more complicated than thinking just of space and time. He could actually think about the movement of space itself. So he could calculate the movement of the universe and famously discovered the expanding universe. Now there's a lot of it said to be said about how we dealt with that. Einstein didn't really like the idea of an expanding universe. He hated the idea of a big bang. And in fact it was real, it was astronomers actually who discovered experimentally that the universe was expanding. And nowadays we will notice we have these beautiful images where you actually can see the remnants of the big bang. So this is a very early baby picture of the universe. And you have this great animation where you just see how, boost up the big bang, yes, there it goes. So this is 14 billion years of cosmic evolution I think in 20 seconds. So these are kind of the time slices of the universe. And you see slowly the universe, the early stars, you see the galaxies, you see the universe forming. And in some sense, again, as I said, we astronomers too, they think of the universe not only as a series of moments, but they think of the full space-time history. In this picture, time is running from the left to right. So this is very early after the big bang and here's a whole series of slices. This is the whole evolution, the whole history of the universe as a big chunk of space-time. In fact, if you think again about the here and now, and I said, you know, thinking about your past, if you think of the past, say, of this moment right now in this lecture and you go all the way back, you will see in some sense, there's some part of the universe that actually has been influencing this. And that's, you know, as far in some sense as you can see to the very beginning. So, you know, it's quite remarkable that, you know, space-time has a beginning. There are lots of philosophical questions about how time could begin. I'm happy to debate this in the Q&A, but I also want to briefly end my lecture by saying something about the end of time, because that's an equally amazing phenomena. Can time end? And to do that, I have to say a few things about black holes. Now, black holes are astonishing objects. They are very spectacular. For a long time, they were only appearing in physics books and science fiction movies. Now, we see them in space-time. You know, they're stars that have exploded and have sucked up all the matter into a singular point. In fact, we see some black holes in the middle of galaxies. It is an actual picture of a galaxy and the things that are spruing out, there are hundreds of thousands of light-years big, are the products coming out of the black hole. But what is a black hole in terms of the curvature again, the geometry of space-time? So one way to think about it, if you think of space-time as this kind of trampoline or this rubber material, it's taking a bowling ball and throwing it on the trampoline, and it's so heavy, it goes all the way. You get an infinite deep well. So in some sense, it's just a hole in space. That's what it is. It's actually a very adequate description of the phenomena. But it's also a very dangerous place because you can fall in the black hole and disappear. But what happens that there is a thing, and I want to try to explain this to you, this is a movie, this is space-time history of the formation of a black hole. Here's the star. The star is collapsing. It's shrinking, shrinking, shrinking. It creates this thing here, which is this singularity. It's all a matter, squeezed into a point. But around the black hole is a kind of no-go zone. This cylinder here, it's a sphere in three dimensions. In this image, it's a circle. That if you are inside the horizon, usually you're here, you're doomed. The gravity is so strong, everything is sucked in. If you're outside, you're fine. But there's a very deep reason, and perhaps if you don't know this, you will be really shocked. Why are you doomed if you're inside the horizon? It feels like, well, haven't you tried enough? Perhaps there's a clever trick to get out of the horizon. But there's no way you can go, and the reason is because time is flowing differently. In all the pictures I showed, time is flowing upwards. We have these slices, think of it as movie images, one after another. In a black hole, the arrow of time switches when you go into the horizon. So time is moving inwards. So that's actually quite amazing, right? Because if you... Okay, I can draw a circle here. I can say it's one meter from me to the center of the... In a black hole, there's like one minute inside the black hole. You just have a finite amount of time. You know, whatever you do, you're doomed, because, you know, there will be an end. It's like you're watching a movie, so there's one minute to go, you know, there will be a dramatic end. Or at least there will be an end title, you know? The movie will be finished. Time really stops here in the middle. It's an amazing consequence of Einstein's theory. It's again something that he really objected to, because what has it been even for time to end? But it's one of the purest things that can happen in space-time. Something that we are totally not used to. So literally the end of time happens in black holes, and actually if you think about this, we know there are black holes there, even in our own galaxy. There are many stellar black holes in our neighborhood. So apparently nature has found a way to end time in kind of a agreeable way. And the last thing I want to bring to you, and then I'll get some time for Q&A, is the magic that happens if all these ideas come together. I just want to give you one indication. So what we have seen? We have seen that it's very natural to combine space and time together. And that's... You could have done that anyhow, but it is a very convenient way to keep track of your own history. Then we have seen that these particles can do crazy stuff, right? For instance, they can be created out of nowhere, and pairs of particles and antiparticles, they combine. This is what quantum mechanics brings. So what happens if you put these two things together? And this is actually the great insight of Stephen Hawking. So Stephen Hawking noticed the following thing. In empty space, space is not empty. Empty space is full of these quantum particles that are created in pairs and annihilate again. What would happen if such a pair would be created very close to the black hole horizon? This was his thought experiment. So here's a pair of particles. It's created. They're supposed to annihilate after a split second, but just by accident. One is behind the horizon, and one is in front of the horizon. So the one behind the horizon can never come back. You have seen this, right? Because time is pushing it away. The flow of time, which is like a river, just takes this red particle and brings it down to the center of the black hole where it gets squeezed. So the green particle is left alone without a partner to annihilate again and has to escape. So Hawking, if you put this cartoon into mathematical equations, you find actually that black holes using the laws of quantum mechanics can radiate particles. They're not black. They shine. It's like a radioactive atom where now and then a particle can escape. And so if you wait long enough, we think that black holes will evaporate. It takes longer than the lifetime of the universe, perhaps, but they will, very, very, very slowly. So this shows that somehow these problems that we have with the end of time might be resolved if we use quantum mechanics. Now, I must say black holes are some of the mysterious objects we know. And a hundred years ago, physicists were all worried about the atom. The atom was mysterious. Everybody knew atoms were there. Matter was made out of atoms. But the laws of physics couldn't construct them. They would actually not behave. Yet they are there. So nature found a way around. Well, it took us literally a hundred years to understand how quantum particles behave. If you think about black holes, on the one hand, from a quantum mechanical point of view, they're extremely complicated, but from a geometrical point of view, they're just a hole in space. So in some sense these two laws of physics have to start talking to each other. And I think that's somehow the struggle of modern physics. Not to the end of the story. I think actually many of us feel that in the end, the one thing that you might think is universal and will be there forever, space and time. It's something that we experience every moment. You don't realize it today. But you were moving through space. You were moving through time. It's here around us. Isn't it remarkable that the most common thing, the thing that we all experience, is in some sense the most mysterious object in physics? So with that thought, I want to finish. Thank you. All right, if anybody has any questions, you can raise your hand. I have a question about what seems to be a paradox in the black hole description that you gave. A black hole does exist in time. It exists. But then you said in the center of the black hole, time stops. So I can't resolve that. Do you explain that? Yes. The thing is, if you think about what is a black hole, I draw this image here, right? So this is the horizon of the black hole. So that clearly exists. So there's this sphere around the singularity. It's the most you will ever see of the black hole. There's something inside that horizon. We don't know exactly what happens. So time stops inside. But outside, it's flowing. So that's your paradox, right? Because if I'm here, I'm looking at this object. It's perfectly stable. It's out there. The moment I cross the horizon, then certainly time is flowing in the other direction and it stops after, say, a minute or two. So your paradox, how can time both stop and something be there permanently? It's because you are thinking, you're still thinking like Newton. You're thinking there's one big universal time. There's a clock that's ticking and that's coordinating everything. It's much more complicated. It depends, you know, what you do, who you are. Time is changing in every part of space. So it's more like, instead of a big clock, you should think of as water running over a landscape. Some of it is running to the left, to the right. So it's this flow of time is very irregular. So it's a little bit like here is a little waterfall where the water is flowing inside. Well, here it's just continuing straight. And I think all of us will be on the outside of the black hole. It could be an unfortunate person who falls in, but the person who's actually here, the physicist who's doing the experiments here, has no way to communicate with us. And that's one of the big insights of relativity theory that your experience of time is something very personal. Where you are, where you're going, how you're moving. So the permanence of the black holes from the outside observer. I had a question about the Hawking radiation. It makes sense to me that the particles can split and one goes in and one goes out. But how does that lead to the evaporation of the black hole? Because it seems to me like mass is being added to the black hole. So the thing is, these particles that are created, they are what I said, virtual particles. So they do not behave. They do not have to allow to obey the standard laws of physics like conservation of energy. So if a particle is created out of nowhere, they can't have both positive energy. So one would typically have positive. The other had negative. Usually they're both like totally weird. So that's why they have the inlet again. If this particle actually wants to survive to infinity, it has to be a real particle. So with positive mass. Then by definition, this particle has negative mass. So a negative mass particle falls into the black hole, but somehow it doesn't add. It subtracts mass. So if you're wondering what exactly is happening here, it looks all pretty dubious. But a great thing, it's all behind the horizon, right? So from the outside, we don't know exactly how the laws of physics manage this. But we do know, since positive mass escaped, that negative mass has had to fall in. Yes, I would like to ask you a question. How does the concept of entropy, the ever-increasing randomness in the universe, how does that affect the time-space continuum? Ah, okay. So I would say that it's a little bit related to the question of the arrow of time. So we have these slices. So you could ask, why is time going upwards? Even if you think of just classical physics, you have all the slices. It's pre-sliced bread, but it's not precise ordered. Why don't turn the whole universe upside down? You can. The laws of physics and the microscopic laws of physics are such that you could, in principle, read them in both directions. So the traditional story here is that from a microscopic point of view, there's the concept of entropy, which basically has to do how generic is a certain state. It's something very special. So the typical example is, you take an egg and it falls on the floor. It smashes in many pieces. The reversed image, the pieces all coming together, floating upwards into a perfect egg, could happen, but again with a very small probability. So probabilities kind of make you go in one direction. And so the traditional theory of physics, it is related to the special state of the universe just after the Big Bang. So the universe kind of picked a direction of the error of time. The whole, basically everything I said before discussing cosmology was all symmetric about the orientation of time. So the whole question of the Feynman diagrams and everything could have easily worked if I would have my diagrams all upside down. Hi. Thank you for a great lecture. Very entertaining and informative. I read what some place in the Feynman diagrams it could be interpreted as the future influencing the present. I was wondering if you could speak about that a little. Yeah, that's kind of complicated. It's related to what's called a propagator that you use in Feynman diagrams. I think one point to realize is that whenever you draw these Feynman diagrams, again, I think about the canonical diagram that we had with essentially the one where we had a flash of light come from one to the other. So the question is what is, you might say, one is the origin, so the flash of light goes off, it travels to the other case. But as I said, the ordering in time of these moments is not obvious. And in fact, they could happen in a place where there's no natural causal relation because it's not a real physical light ray connecting them. So in some sense, if you interpret these diagrams, you might even think that there's a point in the future that travels to a point in the past. And it's exactly these kind of things what made it such a big leap for Feynman to take these rules seriously. Because if you're very naive, start to apply Feynman diagrams, you get into a big mess. And his insight was that if you allow yourself that flexibility, so basically what he found are many diagrams, there are many physical processes. Once where particle A influences B, one by particle B influences A, they can all be captured by this one diagram by thinking of it in many ways embedded in space-time. So I think that's perhaps connecting to this. So one way to say this is there are many ways to reach these diagrams in terms of the casual order. And in some sense the diagram is capturing all of that at once. So that was the extreme simplification. So at that time of the Feynman diagrams, the traditional method was to compute all these different processes one by one. And you get enormous bookkeeping. So that literally were calculations that took weeks. And then Feynman realized, wait a minute, it's one diagram. There are many stories about where he went to a conference where somebody was describing the latest result of months of calculation. He said, what did they do? And within a minute or something he had the right number. Because somehow one diagram can capture all these different interpretations. So because of relativity, you can't ever say that one event comes before another. But then how can you explain cause and effect and one event causing another? So as I said, if there are two, say this event, and you have another event which is here, it has no causal relation because it cannot influence. If it's here, then this will influence this. And this can mean I can send a signal from here to here. But I cannot send a signal from here to here. So there are certain events that are causally related. So an event in this part, in this part this would be the cause of this. This could be the cause of that. But something here and this point, I have no causal relations. Yes? Just say I'm moving. And so that event in a certain, for someone who's moving really fast, doesn't happen before the other event. Yeah, so that was this point, right? That you sometimes think can happen at the same time. Or not, if you have different slides. So synchronicity happening at the same time doesn't make sense. So in some sense these things, which either can happen at the same time or not, can never actually have a causal relation. So that's the solution to it. If they have a causal relation, they cannot be at the same time. Yeah, so I would say what, so all the events which are some sense outside here, they have this ambiguity. So you might think that there's a special role for everything on this plane, because they happen at the same time, but that plane has no significance. All right, well, time moves on for us. So one more hand. Thank you very much for coming.